UPSC NDA I 2024 Mathematics Paper

Concept:The function is an integral of a periodic absolute sine function over an interval of 100$\pi$.Explanation:We are given $varphi(a) = \int\limits_{a}^{a+100\pi} |\sin x| \, dx$.The function $|\sin x|$ has period $\pi$.Over one period $[0,\pi]$, the integral is $\int\limits_{0}^{\pi} \sin x \, dx = [-\cos x]_{0}^{\pi} = 2$.The interval length is $100\pi$, which equals exactly 100 periods of length $\pi$.Since the integrand is periodic, the integral over any full period is the same, regardless of the starting point $a$.Therefore, the total integral is 100 times the integral over one period: $100 \times 2 = 200$.Thus $varphi(a) = 200$.Answer:$200$ (Option D).